# Two representation

1. Jan 8, 2014

### LagrangeEuler

1. The problem statement, all variables and given/known data
$e = \begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}$,
$a =\frac{1}{2} \begin{bmatrix} 1 & -\sqrt{3} \\[0.3em] -\sqrt{3} & -1 \\[0.3em] \end{bmatrix}$.
$b =\frac{1}{2} \begin{bmatrix} 1 & \sqrt{3} \\[0.3em] \sqrt{3} & -1 \\[0.3em] \end{bmatrix}$
$c= \begin{bmatrix} -1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}$
$d=\frac{1}{2} \begin{bmatrix} -1 & \sqrt{3} \\[0.3em] -\sqrt{3} & -1 \\[0.3em] \end{bmatrix}$
$f=\frac{1}{2} \begin{bmatrix} -1 & -\sqrt{3} \\[0.3em] \sqrt{3} & -1 \\[0.3em] \end{bmatrix}$
This is irreducible representation of group $S_3$. \\
Reducible representation of $S_3$ is
$e=d=f = \begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}$
$a =b=c=\frac{1}{2} \begin{bmatrix} -1 & -\sqrt{3} \\[0.3em] -\sqrt{3} & 1 \\[0.3em] \end{bmatrix}$
Why is better to use irreducible then reducible representation in this case and in general?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 8, 2014

### Office_Shredder

Staff Emeritus
What do you mean by "better to use?" You haven't used any representations to do anything.

3. Jan 9, 2014

### LagrangeEuler

In practice one always take some irreducible representation to work with. My question is why?